Modeling Radioactive Decay: Solving an IVP for a Series of Three Elements

[nate9519]

1. The Problem

Construct a mathematical model (system of differential equations) for a radioactive series of 3 elements. X,Y, and Z (Z is a stable element). (Note: W decays into X, X decays into Y, and Y decays into Z). At time zero there are 100e (approx. 271.828) moles of element X. After two hours there are exactly 100 moles each of elements X and Y. Solve the IVP and calculate the number of moles for each of the three elements after i) 1 hour and ii) 5 hours

2. Homework Equations

dw/dt = kW
dx/dt = kW - cX
dy/dt = cX - aY
dz/dt = aY
( k ,c ,and a are all different constants)
x(0) = 100e
x(2) = 100
y(2) = 100 3. Attempt at solution

Im posting this to see if i have my system set up correctly. I am not looking for an answer.

 

[Quantum Defect]

1. The Problem

Construct a mathematical model (system of differential equations) for a radioactive series of 3 elements. X,Y, and Z (Z is a stable element). (Note: W decays into X, X decays into Y, and Y decays into Z). At time zero there are 100e (approx. 271.828) moles of element X. After two hours there are exactly 100 moles each of elements X and Y. Solve the IVP and calculate the number of moles for each of the three elements after i) 1 hour and ii) 5 hours

2. Homework Equations

dw/dt = kW
dx/dt = kW - cX
dy/dt = cX - aY
dz/dt = aY
( k ,c ,and a are all different constants)
x(0) = 100e
x(2) = 100
y(2) = 100


3. Attempt at solution

Im posting this to see if i have my system set up correctly. I am not looking for an answer.

Looks good to me, except for the first equation: dw/dt = -kw, which I suspect is a typo. [Note adding up all of the rates gives you conservation of atoms, d[W+X+Y+Z]/dt = 0, and you won't get this unless the first equation is -kw.]

 

[nate9519]

Looks good to me, except for the first equation: dw/dt = -kw, which I suspect is a typo. [Note adding up all of the rates gives you conservation of atoms, d[W+X+Y+Z]/dt = 0, and you won't get this unless the first equation is -kw.]

would kW in dx/dt need to be negative also? and would dz/dt need to be negative?

 

[epenguin]

No and no. kW is the rate of formation of x from W. dz/dt is the rate of accumulation of z, in this case simply from Y without any further decay.

You really should decide what symbol you are going to use for w, x, y, z, CAPS or lower case. - could that have helped confuse you?